(a) What does it mean to say that a stochastic process is a Brownian motion? Show that, if $\left(W_{t}\right)_{t \geqslant 0}$ is a continuous Gaussian process such that $\mathbb{E}\left(W_{t}\right)=0$ and $\mathbb{E}\left(W_{s} W_{t}\right)=s$ for all $0 \leqslant s \leqslant t$, then $\left(W_{t}\right)_{t \geqslant 0}$ is a Brownian motion.

For the rest of the question, let $\left(W_{t}\right)_{t \geqslant 0}$ be a Brownian motion.

(b) Let $\widehat{W}_{0}=0$ and $\widehat{W}_{t}=t W_{1 / t}$ for $t>0$. Show that $\left(\widehat{W}_{t}\right)_{t \geqslant 0}$ is a Brownian motion. [You may use without proof the Brownian strong law of large numbers: $W_{t} / t \rightarrow 0$ almost surely as $t \rightarrow \infty$.]

(c) Fix constants $c \in \mathbb{R}$ and $T>0$. Show that

$$\mathbb{E}\left[f\left(\left(W_{t}+c t\right)_{0 \leqslant t \leqslant T}\right)\right]=\mathbb{E}\left[\exp \left(c W_{T}-\frac{1}{2} c^{2} T\right) f\left(\left(W_{t}\right)_{0 \leqslant t \leqslant T}\right)\right]$$

for any bounded function $f: C[0, T] \rightarrow \mathbb{R}$ of the form

$$f(\omega)=g\left(\omega\left(t_{1}\right), \ldots, \omega\left(t_{n}\right)\right),$$

for some fixed $g$ and fixed $0<t_{1}<\ldots<t_{n}=T$, where $C[0, T]$ is the space of continuous functions on $[0, T]$. [If you use a general theorem from the lectures, you should prove it.]

(d) Fix constants $x \in \mathbb{R}$ and $T>0$. Show that

$$\mathbb{E}\left[f\left(\left(W_{t}+x\right)_{t \geqslant T}\right)\right]=\mathbb{E}\left[\exp \left((x / T) W_{T}-\frac{1}{2}\left(x^{2} / T\right)\right) f\left(\left(W_{t}\right)_{t \geqslant T}\right)\right]$$

for any bounded function $f: C[T, \infty) \rightarrow \mathbb{R}$. [In this part you may use the Cameron-Martin theorem without proof.]